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Full text of "Treatise On Analysis Vol-Ii"

8    COMPLETE HILBERT ALGEBRAS       369

for all n, and all the AB have the same value y = (et \ e^) ~l. Hence, for all x9 y in
A, we have

(xan | yan) = (y*x \ an a*) = ( j;*x | yen) = y(xen \ yen) ;

since the series with general term (xen \ yen) is absolutely convergent, with
sum (x\y) (15.8.11), it follows that if A is infinite-dimensional then U{(x) is a
Hilbert-Schmidt operator, and the relation (15.8.14.1) is valid. Since A is a
Hilbert space, so is its image under U{ , and to show that this image is the
whole of the Hilbert space JS?2(0 (15.4.8), it is enough to show that £7,(A) is
dense in «^2(0- Now, for each pair, m, n with m ^ w, we have

em A** ' *n A*i = *m(AO(A*i) = emAe1

(1 5.8.1 2(iii)), and since enAe1 = Can, it follows that there exists emn eemAen
such that emnan = am (which implies that emn = y"1ama*)J and clearly
ZmnVp = 0 if p^n. We conclude from this that £mij = t/r(O is the con-
tinuous endomorphism of the Hilbert space I such that Emn • an = am and
Emn ' ap = 0 if p ^ n. Our assertion now follows from the fact that the finite
linear combinations of the Emn are dense in J$f2(0 (15.4.8). The proof is
analogous but simpler when A is finite-dimensional.

We remark that, in all cases, the C/r(In) consist of endomorphisms of the
form U(x) oPn and therefore consist of endomorphisms of rank 1 of I.

(15.8.15)    Under the hypotheses of (15.8.1 4), if there exists an element ^0 in
the center of A, then A Is finite-dimensional. In that case the center of A is Cw,
where u Is the unit element of A.

If c e A belongs to the center of A, then C/^c) is an endomorphism of the
A-module I, hence is a homothety xi-+kx, where A e C (15.8.12). But clearly a
homothety cannot be a Hilbert-Schmidt operator on an infinite-dimensional
Hilbert space, unless it is zero.

(15.8.16)    Let A be a separable complete Hilbert algebra and let ak (keJ)
be the topologically simple Hilbert algebras which are the Hilbert summands
of A (1 5.8.1 3). For each k e J, let \k be a minimal left ideal ofak . Let Vbea non-
degenerate representation (15.5.5) of A in a separable Hilbert space H, such
that V : A i— > 2£ (H) is continuous.

(i) H is the Hilbert sum ofsubspaces Hk (k e 3) stable with respect to F,
such that if Vk is the restriction of V to Hk we have Vk(s) = Qfor all s e ah and
all h 7^ k : so that Vk may be considered as a representation of ak on Hfc .

(ii) If ak is finite-dimensional over C, the representation Vk is the Hilbert
sum of a (finite or infinite) sequence of irreducible representations, each equiv-
alent to the representation Uik ofak (15.8.14).he afc, that is to the case where A is